Braised Carrots with Parmesan Cheese

Ingredients:
1.5 lb carrots
4 tbsp butter (.5 stick)
.25 tspn sugar
3 tbsp freshly grated parmigiano-reggiano cheese
Salt

Equipment:
Non-stick saute pan
Optional: cooking measuring tape

1. Peel the carrots, wash them in cold water, and slice them into 3/8 inch disks. The thin tapered ends can be cut thicker.
2. Choose a saute pan that can contain the carrot rounds spread in a single snug layer, without overlapping. Put in the carrots and butter, and enough water to come .25 inch up the sides. (If you don't have a single pan large enough, use 2 smaller ones, dividing the carrots and butter equally between them.) Turn on the heat to medium. Do not cover the pan.
3. Cook until the water has evaporated, then add salt and the .25 teaspoon sugar.
4. Continue cooking, adding from 2 to 3 tablespoons water as needed. Your goal is to end up with well-browned, wrinkled carrot disks, concentrated in flavor and texture. This will take between 1 and 1.5 hours, durin which time you should keep an eye on them. Stop adding water when they begin to reach the wrinkled browned stage, because there must be no liquid left at the end. (In 30 minutes or so, the carrots will become so reduced in bulk that, if you have been using two pans, you will be able to combine them in a single pan.)
5. When done, add the grated Parmesan, turn the carrots over completely once or twice, transfer them to a warm platter, and serve at once.
Ahead-of-time note: The carrots can be finished entirely in advance, except for the addition of the Parmesan, which you should add only when reheating, just before serving.